Reciprocity and antireciprocity in sound transmission through layered materials including elastic and porous media

Sound transmission through elastic solids and porous media having an elastic frame involves longitudinal and transverse waves. A modelling of sound propagation through layers of these media by transfer matrices, which is equivalent to a representation by analogous multi-port circuits was performed previously. The different ports of the circuits are related to the normal and the tangential strains and velocities on the two faces of the layers. The use of impedance matrices instead of transfer matrices shows that the coupling between the normal and the tangential conjugate variables is antireciprocal, and is reciprocal for conjugate variables having the same nature. Moreover, it can be proven that the transmission loss through strafied materials in contact with the same fluid on the and the rear face is the same in opposite directions of propagation.     

Nonlinear surface waves in media with complex rheology

Weak nonlinear waves in a generalized viscoelastic medium with internal oscillators are considered. The rheological relations contain higher time derivatives of the stresses and strains as well as their tensor products. The method of expansion in a small parameter with the introduction of slow time and a running space coordinate is employed. The first approximation gives wave velocities and relations between the parameters equivalent to the results of an acoustic analysis at elastic wave fronts [1]. The second approximation leads to an evolution equation for the displacement velocity. For this a Fourier-Laplace double integral transformation is used. Reversion to the inverse transforms of the unknown functions leads to an integrodifferential evolution equation, which contains a Hubert transform and is a generalization of the Benjamin-Ono equation of deep water theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–103, September–October, 1990.     

Configuration-dependent friction coefficients and elastic dumbbell rheology

The kinetic theory of nonlinear elastic dumbbells, with bead friction coefficients that depend linearly on the interbead distance, is used to obtain the elongational viscosity and the dumbbell stretching as a function of elongation rate. The results are obtained by solving numerically the “diffusion equation” for the configurational distribution function. No S-shaped curves were found for the elongational viscosity or for the mean-square end-to-end distance. Previous investigators did report S-shaped curves and related “hysteresis” effects. However, their results were based on using mathematical approximations that now appear to be inappropriate.     

Wave fronts in elastic media with temperature dependent properties

This paper deals with the propagation of shock and acceleration wave fronts in elastic media with temperature dependent properties. The partial differential equations governing the evolution of such waves are derived and solved using the method of Charpit. Solutions for wave front propagation in a thermoelastic layer with exponentially temperature dependent properties are obtained.     

The rheology of polymer solution elastic strands in extensional flow

We apply micro-oscillatory cross-slot extensional flow to a semi-dilute poly(ethylene oxide) solution. Micro-particle image velocimetry (μPIV) is used to probe the real local flow field. Extreme flow perturbation is observed, where birefringent strands of extended polymer originate from the stagnation point. This coincides with a large increase in the extensional viscosity. The combination of stagnation point flow and μPIV enables us to investigate directly the stress and strain rates in the strand and so determine the true extensional viscosity of the localised strand alone. The Trouton ratio in the strand is found to be ~4000, amongst the highest values of Trouton ratio ever reported. Consideration of the flow in the exit channels surrounding the highly elastic strand suggests a maximum limit for the pressure drop across the device and the apparent extensional viscosity. This has implications for the understanding of high Deborah number extensional thinning reported in other stagnation point flow situations.     

Quasitransverse shock waves in elastic media with an internal structure

We consider the structure of small-amplitude quasitransverse shock waves in a weakly anisotropic elastic medium which possesses an internal structure generating the wave dispersion. The dispersion is modeled by introducing terms with higher derivatives into the equations of the theory of elasticity, and the dissipation is represented by viscous terms. In one of the two possible cases treated below, the requirement that the discontinuity structure exist leads to a set of admissible discontinuities of complex structure. A considerable part of the shock adiabat consists of a set of short portions and separate points, the number of which increases as the viscosity decreases. This complex set of admissible discontinuities is the general case where the dispersion in the shock-wave structure is sufficiently strong. Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 174–180, March–April, 1999.     

Discontinuous wave interaction with interfaces between anisotropic elastic media

The problem about dynamic interaction of discontinuous waves with interfaces between anisotropic elastic media is considered. To investigate this phenomenon accompanied by formation of reflected and refracted quasi-longitudinal and quasi-shear discontinuous waves, a technique based on joint usage of the zero approximation of the ray theory and method of stereomechanical impact is proposed. It is used for the analysis of the wave front transformation, scattering and focusing. The setup problem solutions can be applied to discovering the most seismically hazardous zones in the earth’s crust, interpretation of geophysical data about geological rock structures and the analysis of the causes of dynamic delamination of layered composite and nanomaterials.     

Effective properties of elastic periodic composite media with fibers

A series solution to obtain the effective properties of some elastic composites media having periodically located heterogeneities is described. The method uses the classical expansion along Neuman series of the solution of the periodic elasticity problem in Fourier space, based on the Green's tensor, and exact expressions of factors depending on the shape of the inclusions. Some properties of convergence of the solution are presented, more specifically concerning the elasticity tensor of the reference medium, showing that the convergence occurs even for empty fibers. The solution is extended for rigid inclusions. A comparison is made with previous exact solutions for a fiber composite made of cylindrical fibers with circular cross-sections and with previous estimates. Different examples are presented for new situations concerning the study of fiber composites: composites with elliptic cross-sections and multi-phase fibrous composites.     

Purely transverse waves in elastic anisotropic media

Formulas are obtained for decompositions of the third- and fourth-rank tensors symmetric in the last two and three indices, respectively, into irreducible parts invariant relative to the orthogonal group of coordinate transformation. The corresponding parts of the decompositions are orthogonal to each other. These decompositions are used to obtain a general representation of the displacement vectors of plane transverse waves in elastic isotropic and anisotropic solids. It is shown that the displacement vectors of transverse waves are second-, third-, and fourth-degree homogeneous polynomials of the wave normal. Special orthotropic materials are found that transmit purely transverse waves for any direction of the wave normal. The eigenmoduli, eigenstates, and engineering constants (bulk moduli, Youngs moduli, Poissons ratios, shear moduli, and Lame constants of the closest isotropic materials) are determined for these materials.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 160–172, January–February, 2005     

On wave propagation in inhomogeneous elastic media

Wave propagation in an inhomogeneous elastic rod or slab is considered. The governing equations are written in a matrix form and transformations are sought which reduce the system to a form associated with the wave equation. Integration of the system is then immediate. It is shown that such reduction may be achieved subject to a function involving the density and elastic parameters of the material adopting certain multi-parameter forms. These parameters are available for fitting to the behaviour of a variety of inhomogeneous elastic materials. A specific initial boundary value problem is solved by utilising the present method.     

Plane SH-waves in harmonically inhomogeneous elastic media

This paper considers the oblique propagation of a plane SH-wave in an inhomogeneous elastic medium whose material properties vary harmonically with a space variable. Assuming a small deviation in the harmonic variation, the method of multiple scales is applied and approximate solutions are obtained for three types of the medium.Effects of oblique propagation of SH-waves on a resonant frequency and on an unstable region of wave propagation are discussed. Effects on the reflection coefficient at a stress-free boundary when the wave propagates in a semi-infinite space are also discussed.     

Coupled elastic waves in uniform isotropic media

This article deals with a certain type of wave in an infinite elastic medium. In contrast to ordinary longitudinal and transverse waves, the amplitude of the type of wave in question depends sinusoidally on the coordinates of a plane which is transverse to the direction of propagation of the wave, i.e., the wave is actually a packet of travelling and stationary waves. Longitudinal waves of this type are always coupled with transverse waves, while transverse waves of the given type may be coupled with longitudinal waves or another transverse wave or may exist as a single wave in the form of a packet containing a travelling wave and a stationary wave. The coupled waves have two phase velocities, which depend on the mechanical properties of the medium, the frequency of vibration, and the wave numbers of the stationary waves. Coupled surface waves in an elastic medium are more general in character than Rayleigh waves; they exhibit dispersion, and they can be used to explain certain seismological observations made during earthquakes—the complete absence of vertical displacements in some cases and the frequent occurrence of horizontal displacements parallel to the wave front. Allowing for the coupling of elastic waves in a layer leads to a more general characteristic equation than the equation obtained in the Rayleigh-Lamb problem. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 19–28, September, 1999.     

Body wave propagation in rotating elastic media

We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation Ω. Wave propagation is shown to be directly related to the Kibel number Ki=ω/Ω, where ω is the wave frequency. Two dispersive waves W₁ and W₂ are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W₁ shows a cutoff frequency ω_c=Ω below which it does not propagate. The case of small angular rotation Ω is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order O(Ki⁻¹).     

Nuclei of strain for bi-material elastic media with sliding interface

The nuclei of strain method, which was previously developed for a perfectly bonded bi-material elastic medium, is modified to obtain the fundamental solutions for the elastic state of a bi-material, isotropic elastic medium with sliding interface. The solutions are conveniently expressed in terms of strength vectors at object and image points which are related by the transmission and reflection matrices given in the paper.     

Effective elastic moduli of granular media

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 91–96, January–February, 1990.     

Inverse scattering for stratified elastic media

The trace method is applied to recover the Lamé parameters λ and μ and the mass density ϱ of a stratified elastic medium.